Deconvolution

In mathematics, deconvolution is an algorithm-based process used to enhance signals from recorded data. Where the recorded data can be modeled as a pure signal that is distorted by a filter (a process known as convolution), deconvolution can be used to restore the original signal.[1] The concept of deconvolution is widely used in the techniques of signal processing and image processing.

Before and after deconvolution of an image of the lunar crater Copernicus using the Richardson-Lucy algorithm.

The foundations for deconvolution and time-series analysis were largely laid by Norbert Wiener of the Massachusetts Institute of Technology in his book Extrapolation, Interpolation, and Smoothing of Stationary Time Series (1949).[2] The book was based on work Wiener had done during World War II but that had been classified at the time. Some of the early attempts to apply these theories were in the fields of weather forecasting and economics.

Description

In general, the objective of deconvolution is to find the solution f of a convolution equation of the form:

${\displaystyle f*g=h\,}$

Usually, h is some recorded signal, and f is some signal that we wish to recover, but has been convolved with a filter or distortion function g, before we recorded it. The function g might represent the transfer function of an instrument or a driving force that was applied to a physical system. If we know g, or at least know the form of g, then we can perform deterministic deconvolution. However, if we do not know g in advance, then we need to estimate it. This is most often done using methods of statistical estimation.

In physical measurements, the situation is usually closer to

${\displaystyle (f*g)+\varepsilon =h\,}$

In this case ε is noise that has entered our recorded signal. If a noisy signal or image is assumed to be noiseless, the statistical estimate of g will be incorrect. In turn, the estimate of ƒ will also be incorrect. The lower the signal-to-noise ratio, the worse the estimate of the deconvolved signal will be. That is the reason why inverse filtering the signal is usually not a good solution. However, if at least some knowledge exists of the type of noise in the data (for example, white noise), the estimate of ƒ can be imporoved through techniques such as Wiener deconvolution.

Deconvolution is usually performed by computing the Fourier transform of the recorded signal h and the distortion function (in general terms, it is known as a transfer function) g. Deconvolution is then performed in the frequency domain (in the absence of noise) using:

${\displaystyle F=H/G\,}$

where F, G, and H are the Fourier transforms of f, g, and h respectively. Finally, the inverse Fourier transform of the function F is taken to find the estimated deconvolved signal f.

Applications

Seismology

The concept of deconvolution had an early application in reflection seismology. In 1950, Enders Robinson was a graduate student at MIT. He worked with others at MIT, such as Norbert Wiener, Norman Levinson, and economist Paul Samuelson, to develop the "convolutional model" of a reflection seismogram. This model assumes that the recorded seismogram s(t) is the convolution of an Earth-reflectivity function e(t) and a seismic wavelet w(t) from a point source, where t represents recording time. Thus, our convolution equation is

${\displaystyle s(t)=(e*w)(t).\,}$

The seismologist is interested in e, which contains information about the Earth's structure. By the convolution theorem, this equation may be Fourier transformed to

${\displaystyle S(\omega )=E(\omega )W(\omega )\,}$

in the frequency domain, where ${\displaystyle \omega }$ is the frequency variable. By assuming that the reflectivity is white, we can assume that the power spectrum of the reflectivity is constant, and that the power spectrum of the seismogram is the spectrum of the wavelet multiplied by that constant. Thus,

${\displaystyle |S(\omega )|\approx k|W(\omega )|.\,}$

If we assume that the wavelet is minimum phase, we can recover it by calculating the minimum phase equivalent of the power spectrum we just found. The reflectivity may be recovered by designing and applying a Wiener filter that shapes the estimated wavelet to a Dirac delta function (i.e., a spike). The result may be seen as a series of scaled, shifted delta functions (although this is not mathematically rigorous):

${\displaystyle e(t)=\sum _{i=1}^{N}r_{i}\delta (t-\tau _{i}),}$

where N is the number of reflection events, ${\displaystyle r_{i}}$ are the reflection coefficients, ${\displaystyle t-\tau _{i}}$ are the reflection times of each event, and ${\displaystyle \delta }$ is the Dirac delta function.

In practice, since we are dealing with noisy, finite bandwidth, finite length, discretely sampled datasets, the above procedure only yields an approximation of the filter required to deconvolve the data. However, by formulating the problem as the solution of a Toeplitz matrix and using Levinson recursion, we can relatively quickly estimate a filter with the smallest mean squared error possible. We can also do deconvolution directly in the frequency domain and get similar results. The technique is closely related to linear prediction.

Optics and other imaging

Example of a deconvolved microscope image.

In optics and imaging, the term "deconvolution" is specifically used to refer to the process of reversing the optical distortion that takes place in an optical microscope, electron microscope, telescope, or other imaging instrument, thus creating clearer images. It is usually done in the digital domain by a software algorithm, as part of a suite of microscope image processing techniques. Deconvolution is also practical to sharpen images that suffer from fast motion or jiggles during capturing. Early Hubble Space Telescope images were distorted by a flawed mirror and were sharpened by deconvolution.

The usual method is to assume that the optical path through the instrument is optically perfect, convolved with a point spread function (PSF), that is, a mathematical function that describes the distortion in terms of the pathway a theoretical point source of light (or other waves) takes through the instrument.[3] Usually, such a point source contributes a small area of fuzziness to the final image. If this function can be determined, it is then a matter of computing its inverse or complementary function, and convolving the acquired image with that. The result is the original, undistorted image.

In practice, finding the true PSF is impossible, and usually an approximation of it is used, theoretically calculated[4] or based on some experimental estimation by using known probes. Real optics may also have different PSFs at different focal and spatial locations, and the PSF may be non-linear. The accuracy of the approximation of the PSF will dictate the final result. Different algorithms can be employed to give better results, at the price of being more computationally intensive. Since the original convolution discards data, some algorithms use additional data acquired at nearby focal points to make up some of the lost information. Regularization in iterative algorithms (as in expectation-maximization algorithms) can be applied to avoid unrealistic solutions.

When the PSF is unknown, it may be possible to deduce it by systematically trying different possible PSFs and assessing whether the image has improved. This procedure is called blind deconvolution.[3] Blind deconvolution is a well-established image restoration technique in astronomy, where the point nature of the objects photographed exposes the PSF thus making it more feasible. It is also used in fluorescence microscopy for image restoration, and in fluorescence spectral imaging for spectral separation of multiple unknown fluorophores. The most common iterative algorithm for the purpose is the Richardson–Lucy deconvolution algorithm; the Wiener deconvolution (and approximations) are the most common non-iterative algorithms.

High Resolution THz image is achieved by deconvolution of the THz image and the mathematically modeled THz PSF. (a) THz image of an integrated circuit (IC) before enhancement; (b) Mathematically modeled THz PSF; (c) High resolution THz image which is achieved as a result of deconvolution of the THz image shown in (a) and the PSF which is shown in (b); (d) High resolution X-ray image confirms the accuracy of the measured values.[5]

For some specific imaging systems such as laser pulsed terahertz systems, PSF can be modeled mathematically.[6] As a result, as shown in the figure, deconvolution of the modeled PSF and the terahertz image can give a higher resolution representation of the terahertz image.

Radio astronomy

When performing image synthesis in radio interferometry, a specific kind of radio astronomy, one step consists of deconvolving the produced image with the "dirty beam", which is a different name for the point spread function. A commonly used method is the CLEAN algorithm.

Absorption spectra

Deconvolution has been applied extensively to absorption spectra.[7] The Van Cittert algorithm (article in German) may be used.[8]

Fourier transform aspects

Deconvolution maps to division in the Fourier co-domain. This allows deconvolution to be easily applied with experimental data that are subject to a Fourier transform. An example is NMR spectroscopy where the data are recorded in the time domain, but analyzed in the frequency domain. Division of the time-domain data by an exponential function has the effect of reducing the width of Lorenzian lines in the frequency domain.

See also

• Convolution
• Bit plane
• Digital filter
• Filter (signal processing)
• Filter design
• Minimum phase
• Independent component analysis
• Wiener deconvolution
• Richardson–Lucy deconvolution
• Digital room correction
• Free deconvolution
• Point spread function
• Deblurring
• Unsharp masking

References

1. O'Haver, T. "Intro to Signal Processing - Deconvolution". University of Maryland at College Park. Retrieved 2007-08-15.
2. Wiener, N. (1964). Extrapolation, Interpolation, and Smoothing of Stationary Time Series. Cambridge, Mass: MIT Press. ISBN 0-262-73005-7.
3. Cheng, P. C. (2006). "The Contrast Formation in Optical Microscopy". In Pawley, J. B. (ed.). Handbook of Biological Confocal Microscopy (3rd ed.). Berlin: Springer. pp. 189–90. ISBN 0-387-25921-X.
4. Nasse, M. J.; Woehl, J. C. (2010). "Realistic modeling of the illumination point spread function in confocal scanning optical microscopy". Journal of the Optical Society of America A. 27 (2): 295–302. Bibcode:2010JOSAA..27..295N. doi:10.1364/JOSAA.27.000295. PMID 20126241.
5. Ahi, Kiarash; Anwar, Mehdi (May 26, 2016). Anwar, Mehdi F; Crowe, Thomas W; Manzur, Tariq (eds.). "Developing terahertz imaging equation and enhancement of the resolution of terahertz images using deconvolution". Proc. SPIE 9856, Terahertz Physics, Devices, and Systems X: Advanced Applications in Industry and Defense, 98560N. Terahertz Physics, Devices, and Systems X: Advanced Applications in Industry and Defense. 9856: 98560N. Bibcode:2016SPIE.9856E..0NA. doi:10.1117/12.2228680.
6. Sung, Shijun (2013). Terahertz Imaging and Remote Sensing Design for Applications in Medical Imaging. UCLA Electronic Theses and Dissertations.
7. Blass, W. E.; Halsey, G. W. (1981). Deconvolution of Absorption Spectra. Academic Press. ISBN 0121046508.
8. Wu, Chengqi; Aissaoui, Idriss; Jacquey, Serge (1994). "Algebraic analysis of the Van Cittert iterative method of deconvolution with a general relaxation factor". J. Opt. Soc. Am. A. 11 (11): 2804–2808. Bibcode:1994JOSAA..11.2804X. doi:10.1364/JOSAA.11.002804.